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They meet at 50 second intervals

To see this....Jun runs   50 * 3  = 150m and Quan runs  50 * 5  = 250m....so the total distance covered by both is 150 + 250  = 400m  = 1 lap

So...if they meet for the 6th time.....this must be 300 seconds after they started

So.....Jun has run  300* 3  = 900 m and is 100m from the starting line

And Quan has run  300 * 5 =  1500 m   and is 100m from the starting line

So....they will walk back 100 m together to the starting line

A*3331 + 231=B*1361 + 1247

By simple iteration, A =100 and B =244, so the smallest positive N will be:

100 x 3,331 + 231 =333,331.

Since the LCM of 3,331 and 1,361 =4,533,491. Therefore:

N =4,533,491D + 333,331, where D =0, 1, 2, 3.......etc.

Let the time until they meet =T
3T + 5T =400
8T = 400
T = 400/8 =50 seconds when both will meet for the first time.
50x3 =150 m run by Jun(clockwise)
50x5 =250 m run by Quan(counterclockwise)
150 x 6 =900/400 =2.25 laps run by Jun after their 6th meet
250 x 6 =1,500/400=3.75 laps run by Quan after their 6th meet
2.25 - 2 x 400 =100m from the starting line when Jun meets Quan for the 6th time(clockwise)
3.73 - 3 x 400 =300m from the starting line when Quan meets Jun(counterclockwise) =400 -300 =100 m to the starting line.
So, the shortest distance from the starting line after their 6th meeting will be 100 meters.

Ofcourse, I agree,  we do love your funny pics Rosala! :)

Very nice Supermanaccz,

I would have answered it using similar triangles but your way is much easier 

The other two y coordinates are the same as the other ones, respectively. 

3,17, so one of the y coordinate is 17

9, -4, so one of the y coordinate is -4

The sum of these two values is 13

I hope this helped, just check over my answer. 

:)

Since CX and BY are both heights of the triangle, we have two ways of finding the area of the triangle: 

1) AB * CX ÷ 2

2) AC * BY ÷ 2

Since both of these are the area of the traingle, therefore the two values must be equal. 

We know the values of AC, BY and AB. 

70 * 60 = CX * 75

CX = 56

The trick with these types of problems is that when you see the height drawn out, try to find the area. 

Sidenote: did you find the value of AY, YC, and BC by yourself, since it wasn't on the image, but it was typed out? 

Thanks hectictar, I appreciate the help, you're a lifesaver! :)

First let's look at the two angles drawn on a unit circle separately:

Now....let's look at each triangle...

Look at the first triangle. Since there are  pi  radians in every triangle

∠ABC   =   pi - pi/2 - θ   =   pi/2 - θ

Look at the second triangle. Since there are  2pi  radians in every circle

∠DEF  =   2pi - (θ + 3pi/2)   =   2pi - 3pi/2 - θ   =   pi/2 - θ

And both triangles have a side of length  1 .  So....

△ABC   is  congruent to  △DEF   by  the Ange-Angle-Side rule. And

side  EF   =   side  BC

cos( θ + 3pi/2 )   =   sin θ

I hope this made sense! 

x^5+ 5x^4 -x - 5     let's factor this....so we have

x^4 ( x + 5) - 1 ( x + 5)      the common factor is  x + 5.....so.....

( x +5) ( x^4 - 1)   

And that is writing the original polynomial as a product of two polynomial factors

AOB  = a + b + x + c  =  180°

ZOB =  c =  75°

XOY = b =   63°

AOY =  a + b  = [ 30 + 63 ]  =  93° 

YOZ = x  =   180  -  30 - 63 - 75  = 12°

thanks so much!

Dan and Paul share some money in the ratio 13:5. 
Dan decides this is unfair so he gives Paul £32 of his share to make the ratio 1:1. 
How much did Paul originally have?

Call Paul's original money, P

Call Dan's original money (13/5)P

So, when Dan gives  Paul £32 their money is equal...so we have

(13/5)P  -  32  = P + 32        add 32 to both sides

(13/5)P  =  P +  64        subtract  P from both sides

(8/5)P  =   64         multiply both sides by (5/8)

P =  64 (5/8)   =  £40

Proof

Paul starts with   £40

Dan starts with (13/5) * 40  =  £104

Paul  ends up with 40 + 32  = £72

Dan ends up with  104 - 32  = £72

So....each has equal money, i.e., a 1: 1 ratio

We have x^2+2x=24.

Subtract 24 to both sides to get: x^2+2x-24=0.

You can factor that out. (x+6)(x-4)=0.

So to satisfy that equation, x would be -6 and 4.

x=4, -6.

I don't think there is a way to delete an account.

Besides, why would you want to delete it? It's great here on the forum. 

Maybe a diagram can do more explaining than I can rambling on ad infinitum.

Thank you! I think I learned something new from this problem! 

Thanks, hectictar......I was "feeling my way," here....but...I like your method better  !!

-3  [ 2  - 5    - 1         3  ] 

            -6    33      -96

      _______________

      2   -11   32      -93 ⇒ remainder 

-5 [  2   - 3      1 ]

            -10     65

      ____________

      2      -13     66 ⇒ remainder

-1  [ 1     3     5  ]

             -1    -2

      __________

       1     2     3 ⇒  remainder

4  [  5    - 1        0        6  ] 

             20       76     304

     __________________

      5     19       76     310 ⇒  remainder

In order:

1

4

2

3

x^2 - 4   =   (x - 2) ( x + 2)

x^2  + 25   =    (x - 5i) (x + 5i)

This is a neat question!! 

This drawing shows the information from the problem:

1s + 5s   =   7t + 11t

6s   =   18t

1s   =   3t               (This is like a unit conversion!)

And...

    AB : BC : CD

=  1s : 7t - 1s : 11t                  Let's "convert"  s  into  t ,  using  1s  =  3t .

=  3t : 7t - 3t : 11t

=  3t : 4t : 11t

=  3 : 4 : 11

Looks like you're correct Angel

See the graph , here :  

https://www.desmos.com/calculator/lsth7dplge

Here are some examples, too....

If a polynomial's degree is odd, and leading coefficient is positive:

For example:  y = x ,   which is   y = 1x¹

degree  =  1 , odd

leading coefficient  =  1 , positive

As the  x  values get smaller, the  y values get smaller.

As the  x  values get bigger, the  y  values get bigger.

--------------------

If its degree is odd, and leading coefficient is negative:

For example:  y = -x ,   which is   y = -1x¹

degree  =  1 , odd

leading coefficient  =  -1 , negative

As the  x  values get smaller, the  y values get bigger.

As the  x  values get bigger, the  y  values get smaller.

--------------------

If its degree is even, and leading coefficient is positive:

For example:  y = x² ,   which is   y = 1x²

degree  =  2 , even

leading coefficient  =  1 , positive

As the  x  values get smaller, or more negative, the  y values get bigger.

As the  x  values get bigger, or more positive, the  y  values get bigger.

--------------------

If its degree is even, and leading coefficient is negative:

For example:  y = -x² ,   which is   y = -1x²

degree  =  2 , even

leading coefficient  =  -1 , negative

As the  x  values get smaller, or more negative, the  y values get smaller.

As the  x  values get bigger, or more positive, the  y  values get smaller.